Lagrange Spaces with (𝛾𝛾𝛾 𝛾𝛾𝛾 -Metric

We study Lagrange spaces with ( 𝛾𝛾 , 𝛾𝛾 )-metric, where 𝛾𝛾 is a cubic metric and 𝛾𝛾 is a 1-form. We obtain fundamental metric tensor, its inverse, Euler-Lagrange equations, semispray coeﬃcients, and canonical nonlinear connection for a Lagrange space endowed with a ( 𝛾𝛾 , 𝛾𝛾 )-metric. Several other properties of such space are also discussed.


Introduction
Finsler spaces endowed with ( -metric were studied by several geometers such as Matsumoto [1,2] and Kitayama et al. [3], and various important applications of such spaces have been observed in physics and relativity theory (cf.[4,5]).e notion of ( -metric was taken to a more general space called Lagrange space and the study was performed by the authors such as Miron [6], Nicolaescu [7,8], and the present authors [9].An -dimensional Lagrange space   = ( (  is said to be endowed with ( -metric if Lagrangian (  is function of (  and (  only, that is, (  = (  (  =   (    being a Riemannian metric and (  =   (  a 1-form.Recently, Pandey and Chaubey [10] discussed Lagrange spaces with ( -metric and obtained several results.ey called a Lagrange space   = ( (  to be endowed with ( metric if Lagrangian (  is a function of (  and (  only, that is, (  = (  where (  is a cubic metric and (  is a 1-form, that is,  3 =   (      and (  =   (  .e paper [10] by Pandey and Chaubey is full of �aws and needs to be recti�ed.e aim of the present paper is to develop a revised and modi�ed theory of Lagrange spaces with ( -metric.
e paper is organized as follows.In Section 2, we de�ne a Lagrange space and discuss some preliminary results required for the discussion of the following sections.It includes the notion of a Lagrange space with ( -metric.
In Section 3, we discuss some properties of a Lagrange space with ( -metric and obtain the expression for the fundamental metric tensor   and its inverse   .In Section 4, we consider the variational problem in Lagrange spaces with ( -metric and obtain various forms of Euler-Lagrange equations.Section 5 deals with the semispray of a Lagrange space with ( -metric.In Section 6, we obtain the coefficients of nonlinear connection in a Lagrange space endowed with ( -metric.Section 7 consists of concluding remarks on the results obtained in the paper.

Preliminaries
Let  be an -dimensional smooth manifold and let  be its tangent bundle.Let ( In the present paper, we study a Lagrange space whose Lagrangian  is a function of   and   only, where =      .
Let us denote this Lagrangian by .us    =    .
e space   =    is called a Lagrange space with  -metric (cf.[10]).Following are some examples of regular Lagrangians with  -metric: i    =  3 +  ii    =  3 +  +  +  2       iii    =  +  2 . ( e Lagrange space determined by the Lagrangian in (8)(iii) is reducible to a Finsler space whereas those determined by the Lagrangians in (8)(i) and (8)(ii) are not so.
For basic notations and terminology related to a Lagrange space, we refer the reader to [11,12].

Fundamental Metric Tensor of 𝐿𝐿 𝑛𝑛 = 𝐿𝑐𝑐𝐿 𝐿𝐿 𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿
If we differentiate (5) partially with respect to   and use the symmetry of   in its indices, we obtain where     =       .
Again differentiating (9) partially with respect to   , using symmetry of    in its indices and simplifying, we �nd ∂ ∂  = 2 −2     − 2 −5      (10) where   =     .Differentiating (6) partially with respect to   , we have Differentiating (11) partially with respect to   , we get us, we have the following.
Proposition 1.In a Lagrange space   with  -metric, the following hold good: where e moments of Lagrangian   are given by In our case, the Lagrangian   is a function of  and  only (vide ( 7)).erefore, we have where   =    = .Using ( 9) and ( 11) in ( 16), we obtain Proof.Utilizing Lemma 6.1.2.1 of [11] for the nonsingular matrix   given by (32) we have the result.
In view of corresponding results obtained by us, these results are erroneous.

Euler-Lagrange Equations
Using ( 7) in (2), we obtain For the Lagrangian  given by ( 7 Since we get From   () = (/  ) − (/)(/  ), we have where (51) For the natural parametrization of the curve     [0, 1] ↦   ()   with respect to the cubic metric   (), (, /) = 1.us, we have the following.eorem 10.In the natural parametrization, the Euler-Lagrange equations of a Lagrange space with (, )-metric are If  is constant on the integral curve  of the Euler-Lagrange equations with natural parametrization, then (52) takes the form us, we have the following.eorem 11.If  is constant along the integral curve of the Euler-Lagrange equations with natural parametrization, then the Euler-Lagrange equations of the Lagrange space with (, )metric are given by (53).
Remarks 12. Pandey and Chaubey [10] obtained the following form of Euler-Lagrange equations: In case of natural parametrization, their result is If  is constant along the integral curve  of the Euler-Lagrange equations (with natural parametrization), the Euler-Lagrange equations obtained by Pandey and Chaubey [10] are given by In view of eorem 9, eorem 10, and eorem 11, all the above discussed results of Pandey and Chaubey [10] are erroneous.

Canonical Semispray
In this section, we obtain the coefficients of the canonical semispray of a Lagrange space with ( -metric. Using ( 7) in (3), we obtain Since  3 =   (      and  =   (  , we have where

Canonical Nonlinear Connection
In this section, we obtain the local coefficients of the canonical nonlinear connection of a Lagrange space with ( metric.Partial differentiation of     =    , with respect to   , yields where

Conclusions
We have developed the theory of Lagrange spaces with (, metric.e paper presents a signi�cant generalization of the theory of Lagrange spaces with (, -metrics (cf.[6][7][8]).e expressions for the geometric objects obtained in the paper may be useful in further work on the spaces under consideration.e importance of the results lies in the study of canonical metrical -connection, curvatures, and torsions in such spaces.e expressions for canonical semispray and nonlinear connection, obtained, respectively, in Section 5 and Section 6, may be applicable in geodesic correspondences between two Lagrange spaces with different (, -metrics on the same underlying manifold.It is a matter of later investigations to look into the aforesaid applications of the results obtained in the paper.
and (      be local coordinates on  and , respectively.A Lagrangian is a function      which is a smooth function on   =    and continuous on the null section.e Lagrangian (  is said to be regular if rank (  (  = , where Geometry e integral of action of the Lagrangian   along a smooth curve       leads to the following Euler-Lagrange equations: is a covariant symmetric tensor called the fundamental metric tensor of the Lagrangian ( .A Lagrange space is a pair   = ( (  (  being a regular Lagrangian whose metric tensor   has constant signature on  .
Remarks 3. e scalars  and  1 appearing in eorem 2 are called the principal invariants of the space   .Differentiating (19) and (20), partially with respect to   and simplifying, we, respectively, have ∂    −2   +  −1   , ∂  1   −1   +  0   ,   ,   2  +  −2     +  −1     +      +  0     .e inverse   of the fundamental metric tensor   of a Lagrange space with (, -metric is given by        . ) us, we have the following.eorem2.In a Lagrange space   with  -metric, the moments of Lagrangian   are given by Proposition 4. e derivatives of the principal invariants of a Lagrange space   with (, -metric are given by     ∂  +   ∂  − .(27) Since  and  are positively homogeneous of degree one in   , by virtue of Euler's theorem on homogeneous functions, we have  ∂  +   ∂  ∂  +   ∂ ∂  +   ∂  +   ∂  ∂  +   ∂ ∂  .eorem 6. e fundamental metric tensor of a Lagrange space with (, -metric is given by (32).e following result gives the expression for the inverse of   .eorem 7.